GENERAL I ARTICLE

The Mathematics of Error Correcting Quantum Codes

K R Parthasarathy is INS A

C V Raman Research

Professor at Indian

Statistical Institute, Delhi.

His interests are quantum

probability, mathematical

foundations of quantum

mechanics and probability

theory. He is the author of

two classic books in

probability theory and one

in quantum stochastic

calculus. He is a Fellow of

all the three academies of

sciences in India and is

also a Fellow of TWAS

(Third World Academy of

Sciences). For his

contribution~ to probabil­

ity theory he was awarded

the Bhatnagar Prize in

1976. For his contributions

to quantum stochastic

analysis he was awarded

the TWAS Prize in 1996

and was conferred an

honorary doctorate by the

Nottingham Trent

University, U Kin

July, 2000.

1. Quantum Probability

K R Parthasarathy

1. Introduction

The mathematical theory of communication of messages through a quantum information channel is based on the following three basic principles:

(i) Messages can be encoded as states of a quantum system with a finite number of levels. As an exam­ple one can think of a single -particle spin system with two levels called spin up and spin down which can be labelled 0 and 1, respectively. A bunch of n such 'independent' systems can be viewed as a single quantum system with 2n levels, each level labelled by a word of length n from the binary alphabet {O, 1}.

(ii) Encoded states can be viewed as inputs of a quan­tum channel and transmitted. However, at the receiving end of the channel the output state can differ from the input state owing to the presence of 'noise' in the channel.

(iii) There is a collection of 'good' states at the input which when transmitted through the channel lead to output states from. which the input states can be reconstructed without any error or possibly with a small error.

The good states obeying property (iii) can then be used to encode messages for transmission through the chan­nel. Thus any reasonable theory of error correcting quantum codes should include a proper identification of

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the good states and also an algorithm for (almost) error­free reconstruction of the input state from a knowledge of the output state. To formulate such a theory it is es­sential to understand the notion of a state in quantum theory. In order to make this presentation reasonably self-contained we give here a lightning introduction to quantum probability.

2. Quantum Probability

An elementary quantum system with n levels is described through an n-dimensional complex vector space of all col umn vectors of the form

Such a column vector is called a ket vector. To any such ket vector one can associate a bra vector

which is a row vector, the bar denoting complex con­jugate. If lu), Iv) are two ket vectors then the matrix product

(ullv)

is a scalar called the scalar product between lu) and Iv) and denoted by (u I v ) . The space en of all such ket vectors together with this scalar product is called an n­

dimensional Hilbert space and denoted by the symbol 1t. Any ket vector lu) can also be viewed as a function u on the set {1,2, ,n} with u(i) = zi,i = 1,2-, ,no

n n

If v(i) = ti "V i then (ulv) = LU(i)v(i) = LZiti. i=l i=l

Sometimes it will be convenient to view the index set {1,2, , n} as an abstract set A of n elements, called an alphabet. If T is an n x n matrix over e then it de­fines a linear transformation or linear operator on the

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Hilbert space by lu) ~ Tlu). Note that (uITlv) is a scalar. The space Mn(C) of all n x n matrices over C is an algebra equipped with an involution T ----+ Tt where the ij-th element of Tt is the complex conjugate of the ji-th element of T T is said to be hermitian if T = Tt. One says that T is nonnegative definite or T ~ 0 in symbols if (uITlu) ~ 0 for every lu). If T = T2 = Tt then T is called a projection (onto its range). If T is a projection so is J - T, J being the identity matrix. The sum of all the diagonal elements of T is called its trace and denoted by Tr T

A nonnegative definite matrix p of unit trace is called a state. A hermitian matrix T is called an observable and for any state p, the scalar quantity Tr pT(= Tr Tp) is called the expectation of the observable T in the state p. Even though T may not be hermitian we still say that Tr pT is the expectation of T in the state p. It is important to note that the map

T ~ Tr pT

from the space Mn{ C) into C has all the features of an averaging procedure:

(i) Trp(aTl + bT2) = aTrpTl + bTrpT2, a, b E C, Tl, T2 E

Mn(C);

(ii) 1fT ~ 0 then Tr pT 2 0;

(iii) TrpJ = Trp = 1.

It is good to take a pause and compare these three prop­erties with what one is familiar with in classical prob­ability: Consider the algebra An of all complex valued random variables on the probability space {l, 2, ,n} equipped with the probability distribution PI, P2, ,Pn, Pi being the probability of the elementary outcome i. The map

n

f -+ E f = ~f(i)pi' i=l

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from An into scalars has the properties

(i') E afl + bf2 = aEfl + bEf2, a, bE C, fI, f2 E An,

(ii') If f(i) 2: 0 V i then E f 2: 0

(iii') E 1 = 1 where 1 also denotes the random variable identically equal to unity.

The central difference between the algebra An of ran­dom variables and the algebra M n( C) of matrices is that multiplication in An is commutative whereas multipli­cation in M n (C) is not. This is the reason why quantum probability is also called noncommutative probability.

The most fundamental theorem concerning hermitian matrices (or observables) is the spectral theorem. Ac­cording to this theorem every hermitian matrix T has the form

k

T = LAiEi, (2.2) i=l

where AI, A2, ,Ak are distinct real scalars and El, E2, k

,Ek are projections satisfying LEi = I, EiEj = 0 if i=l

i i= j. The set {AI, A2, , Ak} is called the spectrum of T, the elements Ai are the eigenvalues of T and (2.2) is called the spectral resolution of T We interpret (2.2) as follows: the observable T assumes values AI, A2, ,Ak and the 'event' that T assumes the value Ai is the pro­jection E i . When the spectral theorem is applied to a state p and one also takes into account the fact that every projection E can be expressed as

d

E = L IUi)(Uil, i=l

where {lUi), i = 1,2, ,d} is any orthonormal basis for the subspace {Iu) I Elu) = Iu)} of all ket vectors fixed by E, it follows that p can be expressed as

n

p = LPiIVi) (ViI, (2.3) i=l

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GENERAL I ARTICLE

where (Pl, P2, ,Pn) is a probability distribution on the set {1,2, ,n} and {IVi), i = 1,2, ,n} is an ortho­normal basis for the Hilbert space en, i.e., (viIVj) = Dij for all i, j = 1,2, ,n (where Dij = 1 if i = j and is 0 if i i= j). Equations (2.2) and (2.3) lead to the following statistical interpretation. In the state p the probabil­ity that the observable T assumes the value Ai is equal to Tr pEi 'if i = 1,2, k and the expectation of T is

k k

equal to LAi Tr pEi = Tr P LAiEi = Tr pT which links i=l i=l

the classical definition of expectation and the quantum theoretic definition.

If lu) is any unit vector in en then lu) (ul is a projec­tion whose range is the one dimensional subspace or ray elu). Such a projection is also a state. According to (2.3) every state p can be expressed as a weighted lin­ear combination of states which are the one dimensional projections IVi)(Vil, where the weights Pi constitute a probability distribution. One says that p is a convex combination or a mixture of pure states IVi) (Vi I. A state of the form 1 V) ( v 1 is called pure because if we split 1 v) ( vi as Iv)(vi = PPl + (1- p)p2 where 0 < P < 1 and Pl and P2 are states then Pl = P2 = 1 v) ( v I. In other words a pure state cannot be split into a mixture of two distinct states. We say that the set of all states is a convex set whose extreme points are precisely one dimensional pro­jections. When a pure state has the form Iv)(vl for some unit vector Iv) it is customary to call the unit vector Iv) (or more precisely the unit ray {Alv), IAI = I}) itself as the pure state. What is actually meant is the projection operator Iv)(vl.

The next fundamental notion from quantum probabil­ity that we need is the combination of several quantum systems into a single system. Suppose H j = enj

, j = 1,2, k are the Hilbert spaces describing quantum systems numbered 1,2, ,k, respectively. We wish to describe all of them together as a single system. To

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this end we introduce the tensor 'product' of the Hilbert spaces 11.1, 11.2, 11. k . If I Uoj) E 11. j is given by

(

Zjl 1 Zj2

Zj~j 1 ~ j ~ k, Zjl E C (2.4)

define their tensor product lUI) ® IU2) ® .. ® IUk) (which is also denoted lUI) IU2) IUk)) to be the column vector

lUI, U2, , Uk) = ( Zi) i = i j i 2 . 'k, (2.5)

where

k (2.6)

and the multiindex i runs through in the lexicographic ordering as in a dictionary. For example, when k = 2, nl = 2, n2 = 3 the lexicographic ordering for the dou­ble index ili2 is 11,12,13,21,22,23 so that

Zl1 Z21 Zl1 Z22 Zl1 Z23 Z12Z21 Z12Z22 Z12Z23

Similarly, (UI,U2' ,ukl = (ull(U21 (ukl = (ull ® (u21 ® ® (ukl = (. ,zi' .). The scalar product be-tween two product vectors lUI, U2, ,Uk) and IVI, V2,

k

Vk) is equal to II (uilvi). All product vectors of the form i=1

(2.5) span the Hilbert space cnln

2···n

k and we denote it by 11.1 ® 1i2 ® ® 11.k. If {I Uil), IUi2), IUinJ} is an orthonormal basis for 1ii then the collection

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is an orthonormal basis for 'H I ® 'H2 ®

If Ai is a matrix of order ni x ni and each Ai is viewed as a linear transformation or an operator in 'Hi then one defines the product linear operator Al ® A2 ® ® Ak in 'HI ® 'H2 ® ® 'Hk by

for all product vectors and extending it linearly to all linear combinations of such product vectors. Such a product operator (or matrix) is well defined and one has

(A 1®A2®· ·®Ak)(BI ®B2®· ·®Bk) = A I B I ®A2B2®

® AkBk, (AI ® A2 ® ® Ak)t = At ® A~ ® . ® At,

= aAI ®A2®· ·®Ak+fJAI ®A2®· ·®Ai-I®Bi®Ai+l®

®Ak.

In particular, if each Ai is hermitian so is their product Al ® A2 ® ® Ak. Similarly if each Ai is unitary so is their product. It is a simple exercise to check that

If Pi is a state in 'Hi Vi then PI ® P2 ® ® Pk is a state in 'HI ® 'H2 ® ® 'Hk with the property that for any product observable Al ® ® Ak its expectation in the product state PI ® P2 ® ® Pk is equal to

Tr PIAl ® P2A2 ®

= n:=l Tr PiAi,

which is the product of the expectation of Ai in the state Pi as i varies from 1 to k. Note that a mixture of two distinct product states is not a product state.

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We now come down to the special case of the Hilbert space H = C 2 of dimension 2. Write

10) = (~), 11) = G)' (2.7)

The ket vectors labelled 10) and 11) constitute an ortho­normal basis for C 2 which is viewed as the Hilbert space of a quantum system with two levels denoted 0 and 1. In physical language 0 may denote spin up and 1 spin down for a two level spin system. If aI, a2, ,ak is a binary sequence, i.e., each ai is either 0 or 1 we write

lala2 . ak) = lal) la2) lak) = lal) ® la2) ® ® lak). (2.8)

As we run through all the words ala2 . ak of length k from the binary alphabet {O, I} we see that (2.8) runs through an orthonormal basis for H ® . . ® H = H®k the tensor product being k-fold. As already mentioned unit vectors can be identified with pure states. Thus we have encoded the set of all binary words of length k as a set of pure states in a quantum system which is a product of k elementary systems each of which is described by h = C 2. A state in C 2 is called a one qubit state, where qubit is an abbreviation for a quantum binary digit. A state in H®k is called a k-qubit state. Thus the pure state lala2 ak) is a k-qubit state of the product type. N ow consider a ket vector of the form

11l.) = L cxala2···ak lala2 . ak) (2.9) al,a2,···,ak

where L Icxala2···ak 12 = 1 (2.10)

al,a2,···,ak

and aI, a2, ,ak vary over {O, I}. Then 1'/1.) defines a pure state which is not a product state. We say that the state In) is entangled. One of the interesting questions of our subject is to define an appropriate measure of this entanglement.

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[tb]

Figure 1.

GENERAL I ARTICLE

l Cha..nnel

input &t a.te p output &t a.te T p

noise

More generally, one can speak of an alphabet A which is la finite set containing, say, N elements. Consider now the Hilbert space eN with an orthonormal basis {Ia), a E N} of ket . vectors in eN Then (eN)®k is the k-fold tensor product of copies of eN which has the orthonormal basis

aI, a2, . ak varying in the alphabet A. Thus words of length k from the alphabet A are encoded as pure states from an orthonormal basis of 'H®k

3. Quantum Channels with Noise

A quantum channel can be viewed as a box which, for each input state of a quantum system, produces an out­put state of another quantum system. See Figure l.

Mathematically speaking, for each input state p on a Hilbert space 'H one has an output state T p on probably some other Hilbert space 'H'. For simplicity we assume that 'H = 'H'. Thus the channel effects a transformation T on the space of all states of a quantum system. Each time a 'signal' in the form of a state p is fed into the channel it is transformed into an output state T p but at different times the transformations T may differ! The nature of channel noise is assumed to be such that T

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belongs to a well-defined class of transformations. In general, the transformation T may be nonlinear. Since our sUQJect is in a state of infancy (just five years old!) we assume that the transformation T is always 'affine linear', i.e.,

(3.11)

for any two sates PI, P2 on 'H and p, q are nonnegative scalars satisfying P + q = 1. An example of such a transformation T is given by

(3.12)

where U is a unitary operator. This is an example of a reversible transformation in the sense that T has an in­verse given by T- I P = ut pU. Such a T transforms pure states into pure states. Physically speaking, the state P undergoes a 'Schrodinger dynamics' for one unit of time with U = e- iH , H being a selfadjoint matrix. Sup­pose there is a bunch of unitary operators UI, U2, ,Uk and one of them is chosen at random with probabilities PhP2, ,Pk respectively and applied to a state p. Then one can say that the output state has the structure

k

Tp = ~PjUjpUJ. j=1

Such a T need not be reversible. It can transform a pure state into a mixed state. More generally, one can consider transformations of the form

where

k

Tp = L:LjpL~, j=1

k

LL}Lj = I. j=1

(3.13)

(3.14)

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Figure 2.

input state p

GENERAL I ARTICLE

Since p is positive semide~nite each LjpL} is positive semidefinite and so is their sum. Furthermore,

k

Tr Tp = L Tr LjpL} j=l

k

Tr (LL~Lj)p j=l

Trp

- 1.

Thus T p is again a state. Transformations of the form (3.13) with the restriction (3.14) occur extensively in the physical literature and are known as completely positive maps. Apparently, one can build (or hope to build) de­vices which implement transformations of this kind. The matrices Lj in (3.13) are said to corrupt the input state p and are therefore called error operators. The noise in the channel is specified by demarcating a class A of ma­trices operating as linear operators in the Hilbert space 'H of the quantum system. It is usually assumed that A is also a linear space, called the space of error operators affecting the channel. We now state the basic hypothesis concerning the operation of the channel in Figure 1. See Figure 2.

Channel output state L LipL~, Li E A

i

noise with error operators from A

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Note that the input state p can be pure but the output state can be mixed! For example, if p = I'll,) (ul the output state '2:Lj I'll,) (niL} is a sum of rank one positive

j

semidefinite operators which need not be a projection.

Suggested Reading

[1] K R Parthasarathy, An Introduction to Quantum Stochastic Calculw,

Birkhauser Verlag, Basel, 1992.

[2] P A Meyer, Quantum Probability for Probabilists, Lecture Notes in

Mathematics, Vol. No. 1538, 2nd edition, Springer Verlag, Berlin, 1995.

[3] G S Vijay and Vis hal Gupta, Quantum Computing, Part-I, Resonance, Vol.

5, No.9, 69-81,2000; Part-II, Resonance, Vol. 5, No. 10,66-72,2000.

[4] A 0 Pittenger, An Introduction to Quantum Computing Algorithms

(Progress in Computer Science and Applied Logic, Vol. 19) Birkhauser

Verlag, Basel, 1999.

[5] M A Nielsen and I L Chuang, Quantum Computation and Quantum

Infurmation, Cambridge University Press, 2000.

Address for Correspondence

K R Parthasarathy

Indian Statistical Institute

7, S J S Sansanwal Marg

New Delhi 110016, India.

Email: krp@isld.ac.in

Wladyslaw Sklodowski

with his three daughters

Maria , Bronislawa and

Helena Maria with her sister Bronislawa

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